An asymptotic theorem for minimal surfaces and existence results for minimal graphs in $H^2 \times R$

TL;DR

This paper establishes a sharp asymptotic theorem for minimal surfaces in hyperbolic space times a line, proving non-existence results and constructing minimal graphs with prescribed boundary conditions in complex unbounded domains.

Contribution

It introduces a new asymptotic theorem for minimal surfaces in H^2×R and constructs minimal vertical graphs over non-convex, unbounded domains with specific boundary data.

Findings

No properly immersed minimal surface with boundary in a narrow slab exists.

Constructed minimal vertical graphs over complex unbounded domains.

Proved a sharp asymptotic boundary behavior for minimal surfaces.

Abstract

In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in $H^2\times R$. As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary $C$ is a Jordan curve homologous to zero in the asymptotic boundary of $ H^2\times R,$ say $\partial_\infty H^2\times R$, such that $C$ is contained in a slab between two horizontal circles of $\partial_\infty H^2\times R$ with width equal to $\pi.$ We construct minimal vertical graphs in $H^2\times R$ over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data. Our admissible unbounded domains $\Om$ in $H^2\times \{0\}$ are non necessarily convex and non necessarily bounded by convex arcs; each component of its boundary is properly embedded with zero, one or two points on its asymptotic boundary, satisfying a further geometric condition.